4.74.7 Use Isosceles and Equilateral TrianglesBell Thinger

Classify each triangle by its sides.

1. 2 cm, 2 cm, 2 cm

ANSWER equilateral

ANSWER isosceles

2. 7 ft, 11 ft, 7 ft

3. 9 m, 8 m, 10 m

ANSWER scalene

4.7

4.7

4.7Example 1

SOLUTION

In DEF, DE ≅ DF . Name two congruent angles.

DE ≅ DF , so by the Base Angles Theorem, E ≅ F.

4.7Guided Practice

Copy and complete each statement.

1. If HG ≅ HK , then ? ≅ ? .

HGK, HKGANSWER

If KHJ ≅ KJH, then ? ≅ ? .2. 2.

ANSWER KH, KJ

4.7

4.7Example 2

Find the measures of P, Q, and R.

The diagram shows that PQR is equilateral. Therefore, by the Corollary to the Base Angles Theorem, PQR is equiangular. So, m P = m Q = m R.

3(m P) = 180o

Triangle Sum Theorem

m P = 60o

Divide each side by 3.

The measures of ∠P, ∠Q, and ∠R are all 60°.ANSWER

4.7Guided Practice

3. Find ST in the triangle at the right.

5 ANSWER

4. Is it possible for an equilateral triangle to have an angle measure other than 60°? Explain.

No; The Triangle Sum Theorem and the fact that the triangle is equilateral guarantees the angles measure 60° because all pairs of angles could be considered base angles of an isosceles triangle.

ANSWER

4.7Example 3

ALGEBRA Find the values of x and y in the diagram.

SOLUTION

STEP 1 Find the value of y. Because KLN is equiangular, it is also equilateral and KN ≅ KL. Therefore, y = 4.

4.7Example 3

STEP 2 Find the value of x. Because LNM ≅ LMN, LN ≅ LM and LMN is isosceles. You also know that LN = 4 because KLN is equilateral.

LN = LM Definition of congruent segments

4 = x + 1 Substitute 4 for LN and x + 1 for LM.

3 = x Subtract 1 from each side.

4.7Example 4

Lifeguard Tower

In the lifeguard tower, PS ≅ QR and QPS ≅ PQR.

QPS ≅ PQR?

a. What congruence postulate can you use to prove that

SOLUTION

Draw and label QPS and PQR so that they do not overlap. You can see that PQ ≅ QP, PS ≅ QR, and ∠QPS ≅ ∠PQR. So, by the SAS ≅ Postulate, QPS ≅ PQR.

a.

4.7Example 4

Lifeguard Tower

In the lifeguard tower, PS ≅ QR and QPS ≅ PQR.

b. Explain why PQT is isosceles.

SOLUTION

b. From part (a), you know that 1 ≅ 2 because corresp. parts of ≅ are ≅. By the Converse of the Base Angles Theorem, PT ≅ QT , and

PQT is isosceles.

4.7Example 4

Lifeguard Tower

In the lifeguard tower, PS ≅ QR and QPS ≅ PQR.

c. Show that PTS ≅ QTR.

SOLUTION

c. You know that PS ≅ QR , and 3 ≅ 4 because corresp. parts of ≅ are ≅. Also, PTS ≅ QTR by the Vertical Angles Congruence Theorem. So, PTS ≅ QTR by the AAS Congruence Theorem.

4.7Exit Slip

Find the value of x.

1.

ANSWER 8

4.7

Find the value of x.

2.

ANSWER 3

Exit Slip

4.7

If the measure of vertex angle of an isosceles triangle is 112°, what are the measures of the base angles?

3.

ANSWER 34°, 34°

Exit Slip

4.7

ANSWER 66 cm

Find the perimeter of triangle.4.

Exit Slip

4.7HomeworkPg 279-282#8, 11, 15, 20, 40Classwork (if you finish HW)

pg 279#12, 13, 16, 21, 38